Yes. You need a rest day.
How often should they be? Well, that's a much more difficult question to answer.
Most people think, "If I take one day off each week that will be enough."
Sometimes this is the case, however if you just spent the last 3 days doing hard running intervals, a couple of days off back-to-back might do you some good.
There's also a chance you just spent the last 7 days doing really low intensity work and keeping the stimulus high (not taking a day off) will push your fitness to the next level.
Anyone who says, "You have to have one day off each week" is a liar. Everyone's body, goals, fitness level, experience, age (the list goes on forever) is different. Sometimes you need more than one day, sometimes it's ok to go months without a day off. (yes, I just said that!)
The main thing with a rest day is knowing how to use them and when to use them to accomplish your end goal.
Just remember, your body will never fully recover until you've taken days off. So if you are going into a day or training period where you want to execute at your best, rest is what the body will need.